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Setting up an iterated integral

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To write $I=\iint_D f(x,y) dxdy$ as an iterated integral, $D$ must be expressed in one of two ways:
  1. $x$ varies from $x_\text{min}$ to $x_\text{max}$, and $y$ varies from $y_\text{min}(x)$ to $y_\text{max}(x)$.

  2. $y$ varies from $y_\text{min}$ to $y_\text{max}$, and $x$ varies from $x_\text{min}(y)$ to $x_\text{max}(y)$.
In the first case, the integral has outer variable $x$ and inner variable $y$: $$I=\int_{x_\text{min}}^{x_\text{max}} \int_{y_\text{min}(x)}^{y_\text{max}(x)} f(x,y) dy dx$$
In the second case, the integral has outer variable $y$ and inner variable $x$: $$I=\int_{y_\text{min}}^{y_\text{max}} \int_{x_\text{min}(y)}^{x_\text{max}(y)} f(x,y) dx dy$$
Usually it is most convenient to choose the outer and inner variable so that the description of the domain is as simple as possible.

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